In 2002, Sotomayor sided with the Bush administration in a case brought to court by the Center for Reproductive Rights, which challenged the ban on government funding to international groups that provided abortion or abortion-related services, also known as "Mexico City Policy."

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"We were very disappointed when the decision came out. We thought we had good constitutional claims," Nancy Northup, president of the advocacy organization, the Center for Reproductive Rights, told ABCNews.com. "Judge Sotomayor and the panel believed the precedent that our claims couldn't go forward."

"Because this classification 'neither proceeds along suspect lines nor infringes fundamental constitutional rights,' it must 'be upheld against equal protection challenge if there is any reasonable state of facts that could provide a rational basis for the classification,'" Sotomayor wrote about her decision.

In another 2004 civil rights ruling, Sotomayor ruled on the side of anti-abortion protestors who sued police officers for brutality outside an abortion clinic. And in a 2006 immigration case, she ruled to grant a Chinese woman extended amnesty on the basis that she faced forced birth control in her home country.

Those decisions have some on the left saying her views need to be made public in confirmation hearings.

"Because we don't know where she stands on the constitutional right to abortion, we think it's important that during confirmation hearings, that senators explore her view... and that she expresses her legal views on the matter," Northup said.

"Discussion about Roe v. Wade will -- and must -- be part of this nomination process," Keenan said in an advocacy message sent to supporters.

Asked if Obama knows for a fact that Sotomayor supports the ruling in Roe v. Wade, White House press secretary Robert Gibbs said that the president does not have a litmus test on abortion.

He said Wednesday that Sotomayor and Obama did not talk about abortion or Roe v. Wade in their meeting last week, but he suggested that the two were in agreement on their views in general, adding that the president and Sotomayor talked about her views on "unenumerated rights in the Constitution and the theory of settled law."

"I know he feels comfortable, generally, with her interpretation of the Constitution being similar to that of his," Gibbs said. "He feels comfortable with where she is."

Legal experts say it is too soon to tell how she may vote on a case that resurrects the Roe v. Wade discussion.

"I don't think she's written anything from which people can draw global conclusions," said Lani Guinier, a professor of law at Harvard University. "She's a very careful judge, so you can't predict how she would [rule on a case] unless she has facts."

Sotomayor's Abortion View a Mystery

"The truth is it's not entirely clear how she would vote," Charmaine Yoest, president of pro-life organization Americans United for Life, told ABCNews.com.

"The thing, is in both of those cases, she upheld Supreme Court precedent," she said, referring to the cases involving the Mexico City Policy and the abortion clinic protestors. "Our concern is once she's elevated she doesn't face the same constraints."

Sotomayor added after that statement: "And... I know this is on tape, and I should never say that, because we don't make law, I know. ... I'm not promoting it, and I'm not advocating it."

Yoest believes the abortion argument should not be in the hands of Supreme Court justices.

"Abortion forces want to put it [abortion issue] beyond reach of [the] American people... [But the] legislators are accountable to people," Yoest said. "She has defined a judge as a super legislator. ... Once she's on court, she's beyond the reach of the American people."

Sotomayor's comments during a 2001 lecture at the University of California-Berkeley that she hopes "a wise Latina woman with the richness of her experiences would more often than not reach a better conclusion than a white male who hasn't lived that life," has also raised concern among anti-abortion groups.

"In public policy, being woman is a code word for being an abortion supporter," Yoest said, questioning, "Does she define being a woman as being pro-abortion?"

The Rev. Rob Schenck, president of the National Clergy Council, admitted in a news conference Tuesday that they know little about the judge's views on abortion, only presuming that her ideology might be aligned with Obama's.

"We hope to be convinced that she may be further from that position than the president knows," Schenck said. "We have people doing exhaustive research."

If confirmed, Sotomayor would be the sixth Catholic on the court. Gibbs said as far as he knows, her religious background was not given any consideration when the president selected her.

Northup said Sotomayor's religious beliefs do not necessarily threaten her organization's agenda.

"People's religious beliefs and their views on the legal protections and what Constitution stands for are not necessarily synonymous," Northup said. "We have many judges who are able to apply the constitutional rulings as the law requires, and their religious beliefs are a separate matter."

Sotomayor's Supporters

Despite the mystery surrounding her legal views on Roe v. Wade, some abortion rights groups are cheering on her nomination.

"What more do women want?" National Organization for Women (NOW)'s President Kim Gandy said in a statement, citing Sotomayor's legal achievements. "We want a swift confirmation in the U.S. Senate."

Another hot button issue facing Sotomayor is same-sex marriage. Like abortion, Sotomayor's stance on the issue is unclear. But gay and lesbian groups hail her nomination.

Her "pattern of decision-making makes us more comfortable," D'Arcy Kemnitz, executive director of the National LGBT Bar Association, told ABCNews.com. "I believe that she is going to look at the matters that come before her with a fair mind. She is going to look at them with an empathetic point of view. She is going to ask the tough questions and be fearless in her determination."

"We applaud President Obama for choosing Judge Sonia Sotomayor to become our nation's next U.S. Supreme Court justice," Joe Solmonese, president of the Human Rights Campaign, said in a statement. "We are confident that Judge Sotomayor has a demonstrated understanding and commitment to protecting the liberty and equality of all Americans."